#503305

order	512
nilpotency class	4
isoclinism	isoclinic class of #499712
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁵

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle, prime radical	C₂³	C₂²×D₈
commutator, Frattini, cosocle	C₂²×C₈	C₂⁴
LCS	G ▷ C₂²×C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₂×C₄×Q₈ ◁ C₂×C₈×Q₈ ◁ G
LECS	G ▷ C₂²×C₈ ▷ C₄ ▷ C₂ ▷ C₁

742 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 44
transitive degree	μ_t(G) = 512

A minimal faithful permutation representation — 4 generators on 44 points:

(2,4)(5,6,9,12)(7,11,8,10)(13,14,20,26)(15,21,29,42)(16,30,27,43)(17,35,18,37)(19,39,36,25)(22,28,44,33)(23,31,24,32)(34,40,38,41);(5,7,9,8)(6,10,12,11)(13,15,20,29)(14,21,26,42)(16,31,27,32)(17,34,18,38)(19,33,36,28)(22,25,44,39)(23,30,24,43)(35,40,37,41);(1,2,3,4)(13,16,32,15,19,34,18,28,20,27,31,29,36,38,17,33)(14,22,35,41,25,42,24,43,26,44,37,40,39,21,23,30);(5,8,9,7)(6,11,12,10)

Polycyclic presentation:

2^9|a^2=ei,b^2=ei,c^2=fh,d^2=e,f^2=hi,g^2=hi,h^2=i,b^a=be,c^a=cf,d^a=de,f^a=fh,g^a=gh,h^a=hi,c^b=cg,f^b=fh,g^b=gh,h^b=hi

Conjugacy classes & character table

80 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 410501);